Moment estimates for Lévy Processes

Abstract

For real Lévy processes $(X_t)_{t \geq 0}$ having no Brownian component with Blumenthal-Getoor index $\beta$, the estimate $E \sup_{s \leq t} |X_s - a_p s|^p \leq C_p t$ for every $t \in [0,1]$ and suitable $a_p \in R$ has been established by Millar for $\beta < p \leq 2$ provided $X_1 \in L^p$. We derive extensions of these estimates to the cases $p < 2$ and $p \leq\beta$.